MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvrex2v Structured version   Visualization version   GIF version

Theorem cbvrex2v 3357
Description: Change bound variables of double restricted universal quantification, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2363. Use the weaker cbvrex2vw 3231 when possible. (Contributed by FL, 2-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
cbvral2v.1 (𝑥 = 𝑧 → (𝜑𝜒))
cbvral2v.2 (𝑦 = 𝑤 → (𝜒𝜓))
Assertion
Ref Expression
cbvrex2v (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
Distinct variable groups:   𝑥,𝐴   𝑧,𝐴   𝑥,𝑦,𝐵   𝑦,𝑧,𝐵   𝑤,𝐵   𝜑,𝑧   𝜓,𝑦   𝜒,𝑥   𝜒,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑤)   𝜓(𝑥,𝑧,𝑤)   𝜒(𝑦,𝑧)   𝐴(𝑦,𝑤)

Proof of Theorem cbvrex2v
StepHypRef Expression
1 cbvral2v.1 . . . 4 (𝑥 = 𝑧 → (𝜑𝜒))
21rexbidv 3170 . . 3 (𝑥 = 𝑧 → (∃𝑦𝐵 𝜑 ↔ ∃𝑦𝐵 𝜒))
32cbvrexv 3353 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑦𝐵 𝜒)
4 cbvral2v.2 . . . 4 (𝑦 = 𝑤 → (𝜒𝜓))
54cbvrexv 3353 . . 3 (∃𝑦𝐵 𝜒 ↔ ∃𝑤𝐵 𝜓)
65rexbii 3086 . 2 (∃𝑧𝐴𝑦𝐵 𝜒 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
73, 6bitri 275 1 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑧𝐴𝑤𝐵 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wrex 3062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-10 2129  ax-11 2146  ax-12 2163  ax-13 2363
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator